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Revista de Matemática Teoría y Aplicaciones
Print version ISSN 1409-2433
Rev. Mat vol.29 n.2 San José Jul./Dec. 2022
http://dx.doi.org/10.15517/rmta.v29i2.50888
Artículo
Control óptimo de orden fraccionario para un modelo de eficacia del tratamiento de la tuberculosis con presencia de VIH/sida y diabetes
Fractional-order optimal control for a model of tuberculosis treatment efficacy in the presence of HIV/aids and diabetes
1University of São Paulo, IME-Rua do Matão, 1010 CEP 05508-090 São Paulo, Brazil; erickdelgadomoya@gmail.com
2University of Antilles, Department of Mathematics and Computer Sciences, LAMIA (EA4540), Pointre-à-Pitre, Guadeloupe, France; alain.pietrus@univ-antilles.fr
En este trabajo, aprovechando las ventajas del uso de las derivadas de orden fraccionario en el sentido de Caputo, presentamos un estudio de control óptimo de la eficacia del tratamiento de la tuberculosis (TB) en presencia del VIH/SIDA y la diabetes. El modelo matemático que es controlado se encuentra en (17) y estudia la relación entre la tuberculosis, el VIH/SIDA y la diabetes con respecto a la eficacia del tratamiento y diferencia en tuberculosis multirresistente (TB-MDR), y tuberculosis extremadamente resistente (TB-XDR). La definición de los controles se centra en evitar la reinfección/reactivación, la TB-MDR y la TB-XDR en las diferentes subpoblaciones (TB únicamente, TB-VIH/SIDA y TBDiabetes). El modelo dividido en subpoblaciones nos permite diferenciar los comportamientos de la transmisión y la resistencia y evaluar los diferentes costos en la aplicación de los controles. Realizamos simulaciones computacionales con datos de la literatura para estudiar nuestro problema de control en un escenario específico. Exploramos el comportamiento del número básico de reproducción variando la tasa de contacto efectivo y los parámetros asociados a la resistencia para diferentes valores de α (orden fraccional). Estudiamos diferentes estrategias de control basadas en la activación de los controles y encontramos que la más efectiva es cuando activamos todos los controles. Con esta estrategia, reducimos el número de casos resistentes, principalmente en la TB-XDR en diabéticos que tiene un fuerte impacto en la dinámica de resistencia y transmisión de la tuberculosis. Además, esta estrategia evita el crecimiento futuro del número de casos resistentes.
Palabras clave: diabetes; VIH/SIDA; derivadas fraccionarias en el sentido de Caputo; problema de control óptimo; tuberculosis.
In this paper, taking advantage of the use of fractional order derivatives in the Caputo sense, we present a study of optimal control of tuberculosis (TB) treatment efficacy in the presence of HIV/AIDS and diabetes. The mathematical model to which control is applied is found in (17) and studies the relationship between TB, HIV/AIDS and diabetes with respect to treatment efficacy and differentiates into multidrug-resistant TB (MDRTB), and extensively drug-resistant TB (XDR-TB). The definition of controls focuses on avoiding reinfection/reactivation, MDR-TB and XDR-TB in the different subpopulations (TB only, TB-HIV/AIDS and TB-Diabetes). The model which is divided into subpopulations allows us to differentiate transmission and resistance behaviors and to evaluate the different costs in the application of controls. We performed computational simulations with literature data to study our control problem in a specific scenario. We explored the behavior of the basic reproduction number by varying the effective contact rate and the parameters associated with the resistance for different values of α (fractional order). We studied different control strategies based on the activation of the controls and found that the most effective is when we activate all the controls. With this strategy, we reduce the number of resistant cases, mainly in XDR-TB in diabetics which has a strong impact on the dynamics of TB resistance and transmission. In addition, this strategy avoids future growth in the number of resistant cases.
Keywords: diabetes; HIV/AIDS; fractional derivatives in the Caputo sense; optimal control problem; tuberculosis.
Agradecimientos
Queremos agradecer profundamente el trabajo de los revisores de la revista por todas sus observaciones.
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Recibido: 26 de Abril de 2022; Revisado: 25 de Mayo de 2022; Aprobado: 22 de Junio de 2022
Este es un artículo publicado en acceso abierto bajo una licencia Creative Commons
